I think 1.Kf2 d5 is OK for White as long as he keeps his distance: 2.Ke2 Ke6 3.Kf2 Kd6 4.Ke2. If Black advances to e5 or f5, White opposes on e3 or f3. If Black comes round the queenside, White shuttles between a2 and b2. There's no way through.

After 1.Kg4, how about 1...d5 and if 2.Kf4 then 2...Ke6. Now White is losing.